DOCSLIB.ORG

Contents 1 Renormalization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Contents 1 Renormalization Quantum Field Theory Chap 7 Renormalization Ling-Fong Li December 21, 2014 Contents 1 Renormalization 1 1.1 Renormalization in 4 Theory . 2 1.1.1 Mass and wavefunction renormalization . 4 1.1.2 Coupling constant renormalization . 6 1.2 BPH renormalization . 8 1.2.1 Power counting . 9 1.2.2 Analysis of divergences . 10 1.3 Regularization . 13 1.3.1 Pauli-Villars Regularization . 13 1.3.2 Dimensional regularization . 18 2 Power counting and Renormalizability 20 2.1 Theories with fermions and scalar fields . 20 2.2 Theories with vector fields . 24 2.3 Composite operator . 27 3 Renormalization group 29 3.1 Renormalization group equation 29 3.2 Effective coupling constant . 31 4 Appendix n-dimensional Integration 34 4.1 n-dimensional “spherical”coordinates . 34 4.1.1 (a) n = 2 .......................................... 34 4.1.2 (b) n = 3 .......................................... 34 4.1.3 (c) n = 4 .......................................... 35 4.1.4 (d) General n ........................................ 36 4.2 Some integrals in dimensional regularization . 36 1 Renormalization Many people who have studied relativistic quantum field theory find the most diffi cult part is the theory of renormalization. The relativistic field theory is full of infinities which need to be taken care of before the theoretical predictions can be compared with experimental measurements. These infinities look formidable at first sight. It is remarkable that over the years consistent ways have been found to make sense of these apparently divergent theories and the results compared successfully with experiments. The theory of renormalization is a prescription which consistently isolates and removes all these infinities from the physically measurable quantities. Note that the need for renomalization is quite general and is not unique to the relativistic field theory. For example, consider an electron moving inside a solid. If the interaction between electron and the lattice of the solid is weak enough, we can use an effective mass m to describe its response to an externally applied force and this effective mass is certainly different from the mass m measured outside the solid where there is no interaction. Thus the electron mass is changed (renormalized) from m to m by the interaction of the electron with the lattice in the solid. In this simple case, both m and m are measurable and hence both are finite. For the relativistic field theory, the situation is the same except for two important differences. First the renormalization due to the interaction is generally infinite (corresponding to the divergent loop diagrams). These infinities, coming from the contribution of high momentum modes are present even for the cases where the interactions are weak. Second, there is no way to switch off the interaction between particles and the quantities in the absence of interaction, bare quantities, are not measurable. Roughly speaking, the program of removing the infinities from physically measurable quantities in relativistic field theory, the renormalization program, involves shu­ ing all the divergences into bare quantities which are not measurable. In other words, we can redefine the unmeasurable quantities to absorb the divergences so that the physically measurable quantities are finite. The renormalized mass which is now finite can only be determined from experimental measurement and can not be predicted from the theory alone. Eventhough the concept of renormalization is quite simple, the actual procedure for carrying out the operation is quite complicated and intimidating. In this chapter, we will give a bare bone of this pro- gram. Note that we need to use some regularization procedure ([?]) to make these divergent quantities finite before we can do mathematically meaningful manipulations. It turns out that not every relativistic field theory will have this property that all divergences can be absorbed into redefinition of few physical parameters. Those which have this property are called renormalizable theories and those which don’t are called unrenormalizable theories. This has became an important criteria for choosing a right theory because we do not really know how to handle the unrenormalizable theory. 1.1 Renormalization in 4 Theory To avoid complicaiton from spin and gauge dependence, we consider a simple example of 4 theory where the Lagrangian is given by, = 0 + I L L L 1 2 2 2 0 4 0 = [(@ ) ] , I = L 2 0 0 0 L 4! 0 2 Here 0 , 0, 0 are usually called the bare or unrenormalized quantities and will be renormalized by the interactions. Feynman rule The vertex and propagator are given by Fig 1 Feynman rule for '4 theory 2 Here p is the momentum carried by the line and is the bare mass term in 0. 0 L (a) 4-momentum conservation at each vertex. (b) Integrate over internal momenta not fixed by momentum conservation (c) For the discussion of renormalization, we do not give propagators for external lines Simple example First consider the two point function (propagator) is defined by 4 ip x i (p) = d xe · 0 T (' (x) ' (0)) 0 h j 0 0 j i Z and has contribution from following graphs for example, We define the term one-particle-irreducible, or 1PI as those graphs which can not be made disconnected by cutting any one internal line. It is clear that we can express a general graph in terms of some collections of 1PI graphs connected by a single line. For expample in the above graphs diagram (a) is an 1PI graph while diagram (b) is not. It is not hard to see that we can attribute all the divergence to the 1PI subgraphs. For example, in daigram (b) the divergences are due to the 1PI in 1-loop. So we can concentrate on 1PI graphs for the discussion of renormalization because in the one particle reducible graph the internal line which can be cut to make the graphs a disconnected one will not be integrated over. It is easy to see that we can write the full propagator in terms of 1PI self energy graph as a geometric series, i i i i (p) = + i p2 + (1) p2 2 + i" p2 2 + i" p2 2 + i" ··· 0 0 0 i = p2 2 (p2) + i" 0 Here p2 contains all IPI self energy graphs. In one-loop we have following divergent 1PI graphs, The 1-loop self energy graph given below, i d4l i i(p) = 0 2 (2)4 l2 2 + i" Z 0 is quadratically divergent. The 1-loop 4-point function are with graph (a) gives the contribution (i )2 d4l i i (p2) = 0 2 (2)4 (l p)2 2 + i" l2 2 + i" Z 0 0 and is logarithmically divergent. Note that in p2 , inside the integral the dependence on the external momentum p is in the combination (p l) in the denominator. This means that if we differentiate p2 with respect to the external momenta p, power of l will increase in denominator and will make the integral more convergent, 2 4 @ 2 1 @ 2 0 d l (l p) p 1 2 (p ) = 2 p (p ) = 2 4 2 ·2 2 2 2 convergent @p 2p @p p (2) [(l p) + i"] l + i" ! Z 0 0 This is true whenever the external momenta ocurrs in company with internal momenta in the denominator. Thus if we expand (p2) in Taylor series, 2 2 (p ) = a0 + a1p + ... the coeffi cients are all derivatives of (p2) with respect to p and the divergences are contained in first few terms in the expansion. So we will use Taylor expansion to separate the divergent quantities from the convergent terms. In our simple case, if we write (p2) = (0) + (~ p2) then then only the first term (0) is divergent and second term (~ p2) which contains all higher derivatives terms is finite. Other 1-loop graphs are either finite or contain the above graphs as subgraphs. 1.1.1 Mass and wavefunction renormalization In 1PI self energy, the expansion in external momentum p will have 2 divergent terms because it is quadrat- ically divergent, 2 2 2 2 2 2 2 (p ) = ( ) + (p )0 ( ) + (~ p ) (2) where 2 is an arbitrary reference point for the Taylor expansion. It is clear that the first term (2) is 2 2 ~ 2 quadratically and the second term 0 ( ) logarithmically divergnet. The 3rd term (p ) is finite and satisfies the conditions, 2 2 2 (~ ) = 0, ~ 0 ( ) = 0 Complete propagator is then i i(p2) = p2 2 (2) (p2 2) 2 (~ p2) 0 0 Suppose we choose 2 such that 2 (2) = 2 mass renormalization (3) 0 then (p2) will have a pole at p2 = 2 . The position of the pole 2 can be interpreted as physical mass 2 2 and 0 is the bare mass which appears in the original Lagrangian. Note that in Eq ( 3) ( ) is divergent. 2 2 2 Thus we require the bare mass 0 to be divergent in such a way that the combination 0 ( ), the 2 physical mass, is finite. In other words, we blame the divergence on the bare mass 0, which is not a measurable quantity and diverges in such a way that the combination with the term from the interaction (2) is finite. This interpretation looks strange and is hard to accept at first sight but it is logically consistent and workable Using the mass renormalization condition in Eq (3) we can write the full propagator as i i(p2) = (p2 2)[1 2 (2)] (~ p2) 0 2 ~ 2 Since 0( ) and (p ) are both of order 0 or higher, we can approximate 2 2 2 2 2 (~ p ) (1 0 )(~ p ) + O ! 0 2 2 ~ 2 2 Note that the term we add 0 (p ) is of order 0 and is negligible in the perturbative expansion in .
Load more

Recommended publications
  • Arxiv:1406.7795V2 [Hep-Ph] 24 Apr 2017 Mass Renormalization in a Toy Model with Spontaneously Broken Symmetry
    UWThPh-2014-16 Mass renormalization in a toy model with spontaneously broken symmetry W. Grimus∗, P.O. Ludl† and L. Nogu´es‡ University of Vienna, Faculty of Physics Boltzmanngasse 5, A–1090 Vienna, Austria April 24, 2017 Abstract We discuss renormalization in a toy model with one fermion field and one real scalar field ϕ, featuring a spontaneously broken discrete symmetry which forbids a fermion mass term and a ϕ3 term in the Lagrangian. We employ a renormalization scheme which uses the MS scheme for the Yukawa and quartic scalar couplings and renormalizes the vacuum expectation value of ϕ by requiring that the one-point function of the shifted field is zero. In this scheme, the tadpole contributions to the fermion and scalar selfenergies are canceled by choice of the renormalization parameter δv of the vacuum expectation value. However, δv and, therefore, the tadpole contributions reenter the scheme via the mass renormalization of the scalar, in which place they are indispensable for obtaining finiteness. We emphasize that the above renormalization scheme provides a clear formulation of the hierarchy problem and allows a straightforward generalization to an arbitrary number of fermion and arXiv:1406.7795v2 [hep-ph] 24 Apr 2017 scalar fields. ∗E-mail: [email protected] †E-mail: [email protected] ‡E-mail: [email protected] 1 1 Introduction Our toy model is described by the Lagrangian 1 1 = iχ¯ γµ∂ χ + y χT C−1χ ϕ + H.c. + (∂ ϕ)(∂µϕ) V (ϕ) (1) L L µ L 2 L L 2 µ − with the scalar potential 1 1 V (ϕ)= µ2ϕ2 + λϕ4.
    [Show full text]
  • A Chiral Symmetric Calculation of Pion-Nucleon Scattering
    . s.-q3 4 4t' A Chiral Symmetric Calculation of Pion-Nucleon Scattering UNeOo M Sc (Rangoon University) A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Adelaide Department of Physics and Mathematical Physics University of Adelaide South Australia February 1993 Aword""l,. ng3 Contents List of Figures v List of Tables IX Abstract xt Acknowledgements xll Declaration XIV 1 Introduction 1 2 Chiral Symmetry in Nuclear Physics 7 2.1 Introduction . 7 2.2 Pseudoscalar and Pseudovector pion-nucleon interactions 8 2.2.1 The S-wave interaction 8 2.3 Soft-pion Theorems L2 2.3.1 Partially Conserved Axial Current(PCAC) T2 2.3.2 Adler's Consistency condition l4 2.4 The Sigma Model 16 2.4.I The Linear Sigma, Mocìel 16 2.4.2 Broken Symmetry Mode 18 2.4.3 Correction to the S-wave amplitude . 19 2.4.4 The Non-linear sigma model 2.5 Summary 3 Relativistic Two-Body Propagators 3.1 Introduction . 3.2 Bethe-Salpeter Equation 3.3 Relativistic Two Particle Propagators 3.3.1 Blankenbecler-Sugar Method 3.3.2 Instantaneous-interaction approximation 3.4 Short Range Structure in the Propagators 3.5 Smooth Propagators 3.6 The One Body Limit 3.7 One Body Limit, the R factor 3.8 Application to the zr.lú system 3.9 Summary 4 The Formalisrn 4.r Introduction - 4.2 Interactions in the Cloudy Bag Model . 4.2.1 Vertex Functions 4.2.2 The Weinberg-Tomozawa Term 4.3 Higher order graphs 4.3.1 The Cross Box Interaction 4.3.2 Loop diagram 4.3.3 The Chiral Partner of the Sigma Diagram 4.3.4 Sigma like diagrams 4.3.5 The Contact Interaction 4.3.6 The Interaction for Diagram h .
    [Show full text]
  • Strongly Coupled Fermions on the Lattice
    Syracuse University SURFACE Dissertations - ALL SURFACE December 2019 Strongly coupled fermions on the lattice Nouman Tariq Butt Syracuse University Follow this and additional works at: https://surface.syr.edu/etd Part of the Physical Sciences and Mathematics Commons Recommended Citation Butt, Nouman Tariq, "Strongly coupled fermions on the lattice" (2019). Dissertations - ALL. 1113. https://surface.syr.edu/etd/1113 This Dissertation is brought to you for free and open access by the SURFACE at SURFACE. It has been accepted for inclusion in Dissertations - ALL by an authorized administrator of SURFACE. For more information, please contact [email protected]. Abstract Since its inception with the pioneering work of Ken Wilson, lattice field theory has come a long way. Lattice formulations have enabled us to probe the non-perturbative structure of theories such as QCD and have also helped in exploring the phase structure and classification of phase transitions in a variety of other strongly coupled theories of interest to both high energy and condensed matter theorists. The lattice approach to QCD has led to an understanding of quark confinement, chiral sym- metry breaking and hadronic physics. Correlation functions of hadronic operators and scattering matrix of hadronic states can be calculated in terms of fundamental quark and gluon degrees of freedom. Since lat- tice QCD is the only well-understood method for studying the low-energy regime of QCD, it can provide a solid foundation for the understanding of nucleonic structure and interaction directly from QCD. Despite these successes problems remain. In particular, the study of chiral gauge theories on the lattice is an out- standing problem of great importance owing to its theoretical implications and for its relevance to the electroweak sector of the Standard Model.
    [Show full text]
  • Optimized Probes of the CP Nature of the Top Quark Yukawa Coupling at Hadron Colliders
    S S symmetry Article Optimized Probes of the CP Nature of the Top Quark Yukawa Coupling at Hadron Colliders Darius A. Faroughy 1, Blaž Bortolato 2, Jernej F. Kamenik 2,3,* , Nejc Košnik 2,3 and Aleks Smolkoviˇc 2 1 Physik-Institut, Universitat Zurich, CH-8057 Zurich, Switzerland; [email protected] 2 Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia; [email protected] (B.B.); [email protected] (N.K.); [email protected] (A.S.) 3 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia * Correspondence: [email protected] Abstract: We summarize our recent proposals for probing the CP-odd ik˜t¯g5th interaction at the LHC and its projected upgrades directly using associated on-shell Higgs boson and top quark or top quark pair production. We first recount how to construct a CP-odd observable based on top quark polarization in Wb ! th scattering with optimal linear sensitivity to k˜. For the corresponding hadronic process pp ! thj we then present a method of extracting the phase-space dependent weight function that allows to retain close to optimal sensitivity to k˜. For the case of top quark pair production in association with the Higgs boson, pp ! tth¯ , with semileptonically decaying tops, we instead show how one can construct manifestly CP-odd observables that rely solely on measuring the momenta of the Higgs boson and the leptons and b-jets from the decaying tops without having to distinguish the charge of the b-jets. Finally, we introduce machine learning (ML) and non-ML techniques to study the phase-space optimization of such CP-odd observables.
    [Show full text]
  • Yukawa Potential Orbital Energy: Its Relation to Orbital Mean Motion As Well to the Graviton Mediating the Interaction in Celestial Bodies
    Hindawi Advances in Mathematical Physics Volume 2019, Article ID 6765827, 10 pages https://doi.org/10.1155/2019/6765827 Research Article Yukawa Potential Orbital Energy: Its Relation to Orbital Mean Motion as well to the Graviton Mediating the Interaction in Celestial Bodies Connor Martz,1 Sheldon Van Middelkoop,2 Ioannis Gkigkitzis,3 Ioannis Haranas ,4 and Ilias Kotsireas4 1 University of Waterloo, Department of Physics and Astronomy, Waterloo, ON, N2L-3G1, Canada 2University of Western Ontario, Department of Physics and Astronomy, London, ON N6A-3K7, Canada 3NOVA, Department of Mathematics, 8333 Little River Turnpike, Annandale, VA 22003, USA 4Wilfrid Laurier University, Department of Physics and Computer Science, Waterloo, ON, N2L-3C5, Canada Correspondence should be addressed to Ioannis Haranas; [email protected] Received 25 September 2018; Revised 25 November 2018; Accepted 4 December 2018; Published 1 January 2019 Academic Editor: Eugen Radu Copyright © 2019 Connor Martz et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Research on gravitational theories involves several contemporary modifed models that predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. In this paper we consider a Yukawa potential and we calculate the time rate of change of the orbital energy as a function of the orbital mean motion for circular and elliptical orbits. In both cases we fnd that there is a logarithmic dependence of the orbital energy on the mean motion. Using that, we derive an expression for the mean motion as a function of the Yukawa orbital energy, as well as specifc Yukawa potential parameters.
    [Show full text]
  • Topics in Two-Dimensional Heterotic and Minimal Supersymmetric Sigma Models
    Topics in Two-dimensional Heterotic and Minimal Supersymmetric Sigma Models A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jin Chen IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Prof. Arkady Vainshtein May, 2016 c Jin Chen 2016 ALL RIGHTS RESERVED Acknowledgements I am deeply grateful to my advisor, Arkady Vainshtein, for his step by step guidance, constant encouragement and generous support throughout these years at University of Minnesota. Arkady has been a knowledgeable, passionate, and also patient advisor, mentor and friend to me. He teaches me physics knowledge and ways of thinking, fosters my independent study, as well as tolerates my capriciousness and digressions on research. Besides conduct of research, I also got greatly enjoyed and influenced by Arkady's optimistic and humorous personality during countless discussions and debates with him. I also want to express my special appreciation and thanks to Misha Shifman, who co- advised me and provided many insightful ideas on research and precious suggestions on my physics career during these years. Misha introduced me the topic on heterotic sigma model years ago and led me to this long but fascinating journey, while at that time I was not able to appreciate the beauty of the topic yet. I would like to thank Sasha Voronov. Arkady introduced me to Sasha. I learned from him how to calculate characteristic classes of the Grassmannian manifolds which was a key step in my research. I thank Sasha, Ron Poling and Misha Voloshin for serving on my thesis committee as well.
    [Show full text]
  • 1. Introduction
    TWO-QUARK BOUND STATES: MESON SPECTRA AND REGGE TRAJECTORIES 1 2 G. Ganbold • t and G.V. Efimov1 (1) Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna (2) lnstitute of Physics and Technology, MAS, Ulaanbaatar 1' E-mail: ganbold@thsunl .jinr. ru Abstract We consider simple relativistic quantum field models with the Yukawa interaction of quarks and gluons. In the presence of the analytic confinement there exist bound states of quarks and gluons at relatively weak coupling. By using a minimal set of free parameters (the quark masses and the confinement scale) and involving the mass-dependent coupling constant of QCD we satisfactorily explain the observed meson spectra and the asymptotically linear Regge trajectories. 1. Introduction The conventional theoretical description of colorless hadrons within the QCD implies they are bound states of quarks and gluons considered under the color confinement. A realisation of the confinement is developed in [l, 2] by using an assumption that QCD vacuum is realized by the self-dual homogeneous vacuum gluon field. Accordingly, this can lead to the quark and gluon confinement as well as a necessary chiral symmetry breaking. Hereby, propagators of quarks and gluons in this field are entire analytic fonctions in the p2-complex plane, i.e. the analytic confinement takes place. On the other hand, there exists a prejudice to the idea of the analytic confinement (for example, [3]). Therefore, it seems reasonable first to consider simple quantum field models in order to investigate only qualitative effects. In our earlier paper [4] we considered simple scalar­ field models to clarify the "pure role" of the analytic confinement in formation of the hadron bound states.
    [Show full text]
  • Remark on the QCD-Electroweak Phase Transition in a Supercooled Universe
    Remark on the QCD-electroweak phase transition in a supercooled Universe Dietrich B¨odeker1 Fakult¨at f¨ur Physik, Universit¨at Bielefeld, 33501 Bielefeld, Germany Abstract In extensions of the Standard Model with no dimensionful parameters the elec- troweak phase transition can be delayed to temperatures of order 100 MeV. Then the chiral phase transition of QCD could proceed with 6 massless quarks. The top- quark condensate destabilizes the Higgs potential through the Yukawa interaction, triggering the electroweak transition. Based on the symmetries of massless QCD, it has been argued that the chiral phase transition is first order. We point out that the top-Higgs Yukawa interaction is non-perturbatively large at the QCD scale, and that its effect on the chiral phase transition may not be negligible, violating some of the symmetries of massless QCD. A symmetry breaking top-quark condensation would then happen in a second-order phase transition, a symmetry-breaking first-order transition could occur in the light-quark sector. First-order phase transitions are interesting for cosmology because they can leave ob- servable relics such as the baryon asymmetry of the universe [1,2], gravitational waves [3], or large scale magnetic fields [4]. In the Standard Model there is neither an electroweak, nor a QCD, or quark-hadron phase transition: both electroweak symmetry [5, 6] and the approximate symmetries of QCD get broken in a smooth crossover [7, 8] However, in nearly scale-invariant extensions of the Standard Model there can be a first-order electroweak phase transition, and, furthermore, it can have very peculiar fea- tures: The decay rate of the metastable high-temperature phase can be tiny down to very low temperatures, so that the universe supercools well below the critical tempera- ture [9–12].
    [Show full text]
  • The Higgs As a Pseudo-Goldstone Boson
    Universidad Autónoma de Madrid Facultad de Ciencias Departamento de Física Teórica The Higgs as a pseudo-Goldstone boson Memoria de Tesis Doctoral realizada por Sara Saa Espina y presentada ante el Departamento de Física Teórica de la Universidad Autónoma de Madrid para la obtención del Título de Doctora en Física Teórica. Tesis Doctoral dirigida por M. Belén Gavela Legazpi, Catedrática del Departamento de Física Teórica de la Universidad Autónoma de Madrid. Madrid, 27 de junio de 2017 Contents Purpose and motivation 1 Objetivo y motivación 4 I Foundations 9 1 The Standard Model (SM) 11 1.1 Gauge fields and fermions 11 1.2 Renormalizability and unitarity 14 2 Symmetry and spontaneous breaking 17 2.1 Symmetry types and breakings 17 2.2 Spontaneous breaking of a global symmetry 18 2.2.1 Goldstone theorem 20 2.2.2 Spontaneous breaking of the chiral symmetry in QCD 21 2.3 Spontaneous breaking of a gauge symmetry 26 3 The Higgs of the SM 29 3.1 The EWSB mechanism 29 3.2 SM Higgs boson phenomenology at the LHC: production and decay 32 3.3 The Higgs boson from experiment 34 3.4 Triviality and Stability 38 4 Having a light Higgs 41 4.1 Naturalness and the “Hierarchy Problem” 41 4.2 Higgsless EWSB 44 4.3 The Higgs as a pseudo-GB 45 II Higgs Effective Field Theory (HEFT) 49 5 Effective Lagrangians 51 5.1 Generalities of EFTs 51 5.2 The Chiral Effective Lagrangian 52 5.3 Including a light Higgs 54 5.4 Linear vs.
    [Show full text]
  • Massive Graviton Geons
    PHYSICAL REVIEW D 97, 044005 (2018) Massive graviton geons † ‡ Katsuki Aoki,1,* Kei-ichi Maeda,1, Yosuke Misonoh,1, and Hirotada Okawa2,1,§ 1Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan 2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Received 24 October 2017; published 5 February 2018) We find vacuum solutions such that massive gravitons are confined in a local spacetime region by their gravitational energy in asymptotically flat spacetimes in the context of the bigravity theory. We call such self-gravitating objects massive graviton geons. The basic equations can be reduced to the Schrödinger- Poisson equations with the tensor “wave function” in the Newtonian limit. We obtain a nonspherically symmetric solution with j ¼ 2, l ¼ 0 as well as a spherically symmetric solution with j ¼ 0, l ¼ 2 in this system where j is the total angular momentum quantum number and l is the orbital angular momentum quantum number, respectively. The energy eigenvalue of the Schrödinger equation in the nonspherical solution is smaller than that in the spherical solution. We then study the perturbative stability of the spherical solution and find that there is an unstable mode in the quadrupole mode perturbations which may be interpreted as the transition mode to the nonspherical solution. The results suggest that the nonspherically symmetric solution is the ground state of the massive graviton geon. The massive graviton geons may decay in time due to emissions of gravitational waves but this timescale can be quite long when the massive gravitons are nonrelativistic and then the geons can be long-lived.
    [Show full text]
  • Search for Dynamical Symmetry Breaking Physics by Using Top Quark
    April 1995 Search for Dynamical Symmetry Breaking Physics by Using Top Quark T.Asaka, N.Maekawa, T.Moroi, Y.Shobuda, and Y.Sumino Department of Physics, Tohoku University Sendai, 980 Japan Abstract We report first results from the investigation on. probing dynamical mech- anism of the electroweak symmetry breaking using top quark. We consider the case where the top quark mass originates from a fermion-antifermion pair condensate, which necessitates (i)strong interaction to cause condensation, and (ii)4-fermi interaction to give top mass. From the observed top mass and the unitarity constraint, we obtain, for the 4-fermi interaction (ii), lower bounds for its strength and upper bounds for its intrinsic new mass scale as we vary the type of the strong interaction (i). Bethe-Salpeter and Schwinger-Dyson equations are solved numerically to study the dynamical symmetry breaking effect semi-quantitatively. — 108 1. Introduction The SU(2) x U(l) gauge theory for describing elcctroweak interactions are very suc- cessful both theoretically and experimentally. However all the experimental tests has been done only for its gauge part. We know that weak bosons and various fermions have their own masses. Because these mass terms are forbidden by gauge symmetry, a mechanism of symmetry breaking is needed, but we have few knowledge about this mechanism so far. Two possibilities have been considered for a symmetry breaking mechanism. In the so- called standard model (SM), symmetry breaking sector consists of an SU(2) gauge doublet of elementary scalar field (Higgs field). By the non-zero vacuum expectation value of this field, gauge symmetry is broken.
    [Show full text]
  • SLAC-PUB-1525 January 1975 (T) QUARK CONFINEMENT, RISING
    SLAC-PUB-1525 January 1975 (T) - -h QUARK CONFINEMENT, RISING TRAJECTORIES AND ASYMPTOTIC BEHAVIdR OF FORM FACTORS* Harald J. W. Muller-Kirstenf Stanford Linear Accelerator Center‘ Stanford University, Stanford, California 94305 r ABSTRACT Several long range force models of quark confinement are considered with particular emphasis on gluon exchange analogous to linear and har- monic potentials. First the nonrelativistic case is discussed and the con- nection between the rise of the Regge trajectories and the power of the potential is derived. Then semirelativistic and relativistic equations are considered. It is shown that in each case the rising trajectories result from large quark-gluon coupling constants. It is also shown that an asymptotic power decrease of bound-state form factors follows only if the interaction contains an additional Coulomb- or Yukawa-like part, and only in the space-like or the time-like domain. The significance of infrared cutoffs is examined, violations of unitarity are pointed out and the behavior in the static limit is discussed. (Submitted for publication. ) *Work supported in part by the U. S. Atomic Energy Commission. f-Max Kade Foundation Fellow. On leave from University of Trier- Kaiserslautern, Kaiserslautern, Germany. Address after February 1, 1975: Fachbereich Physik, UniversitZit Trier-Kaiserslautern, 675 Kaiserslautern, Germany. I 2- 1. Introduction Gsnsiderable experimental evidence has accumulated in recent years in support of the idea that hadrons are composite. Also the most popular model of composite hadrons, the quark model, has been successful in explaining several high energy phenomena. Yet all attempts to isolate and observe the quarks have failed to date.
    [Show full text]